how do I prove that there are infinite primes that have the form xk+y, y<x, and share no common factors?

Whoa!!!
That is too complicated I believe the proof use's funtional analysis. It was first proven by Dirichelt (my avatar ).

In fact in the book "Introduction to Theory of Numbers" by Hardy and Wright. Which contains elementary,algebraic and analytic number theory; proves all theorems in the book (even the prime number theorem) except this theorem! That is how complicated it is.

Lemma: The product of integers of the form is again an integer of the form .

Proof:: If and then, . Q.E.D.

Theorem: There are infinitely many primes of form .

Proof: Assume there are finitely many primes of form call them . Then form an integer,
Prime factorize as,
Note that because it is odd. Thus, for any .
We cannot have that all the primes factors of are of the form for that would imply that has form , which is does not because by the lemma. Thus one of its factors has form . Because an odd prime has one of two forms . But then thus, thus, , because is found among thus an impossibility. Q.E.D.